How to merge vertices without disfiguring mesh?
There are a couple ways you can solve problems like this. To fix those verts that are breaking the curvature of your mesh, here's what I would do:


Set the pivot point to 3D cursor, and move your 3D cursor to the center of your teacup.
Duplicate one of the vertices that is part of the same edge loop as the distorted vertex.
Turn on snap, making sure that it's set to Vertex, Active, and Rotation is enabled.
Rotate the duplicated vertices while snapping to one of the other vertices in line with where the distorted vertex should be.
Snap the distorted vertex to your new vertex.
Select everything and remove duplicate vertices.




You can basically use the same technique to patch gaps in this kind of radial geometry. Duplicate geometry that matches the hole you need to plug, and rotate it into place using vertex snap. I deselect and reselect a vertex here to make it the active vertex, and use that to help guide my snapped rotation about the z-axis.